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This question is related to an exercise on Huybrechts, Complex geometry. I want to calculate the algebraic dimension of the following complex manifolds: $\mathbb{P}^1$ , $\mathbb{P}^n$, and $\mathbb{C}/\mathbb{Z}+i\mathbb{Z}$.

The algebraic dimension is defined as the trascendence degree of the function field of each complex manifold $X$: $a(X)=\text{trdeg}_\mathbb{C} K(X)$. For completion, the function field is the set of global meromorphic functions on the complex manifold.

I have very little understanding of the tools I can use to determine these function fields, since I lack some of the algebraic background generally assumed for studying algebraic and complex geometry.

Any hint or a general direction would suffice.

topolosaurus
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  • What did you try? At least say for $\mathbb P^1$? – Arctic Char Nov 03 '21 at 11:18
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    For the specific case of $\mathbb{P}^1$ I can try and interpret this as the extended complex plane. I know that the only entire/global holomorphic functions on the plane are constants $\simeq \mathbb{C}$. I suppose that I have the degree of freedom of choosing how many poles I want my meromorphic function to have, the degree of these poles, and also to move them around, but I really struggle putting this into algebraic terms. – topolosaurus Nov 03 '21 at 11:36
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    I have been able to show that every rational function $f(z_0:z_1) = \frac{a z_0 + b z_1}{c z_0 + d_z1}$ is a meromorphic function on $\mathbb{P}^1$ (with an argument that extends to $\mathbb{P}^n$. The converse is still out of reach for me. – topolosaurus Nov 03 '21 at 14:59

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By Chow-Serre's wonderful GAGA principle the field of meromorphic functions on a projective algebraic variety $X$ coincides with the field $\operatorname {Rat}(X)$ of its rational functions: see here.
And the transcendance degree over $\mathbb C$ of that field $\operatorname {Rat}(X)$ is equal to the dimension of $X$: actually in older books this was the definition of "dimension", before the newer definition of "dimension" as Krull dimension became the norm.
As a consequence the respective dimensions of $\mathbb{P}^1$ , $\mathbb{P}^n$, and $\mathbb{C}/\mathbb{Z}+i\mathbb{Z}$ are $1,n$ and $1$.

Georges Elencwajg
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  • Thank you for answering a somewhat old question! Still bugged me to these days. – topolosaurus Apr 04 '22 at 09:29
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    You are welcome, dear topolosaurus: this is your reward for posing such well-written and interesting questions on this site. Your handle is quite amusing: I too sometimes use the signature Professosaurus Georgius Elencwajgus when writing to some younger and more agile colleagues... – Georges Elencwajg Apr 04 '22 at 09:42